(Very ample)
Let \(X\) be a noetherian scheme over a field \(k\), let \(D\) be a Cartier divisor, and put \(\mcl=\OO_X(D)\). The divisor \(D\) is very ample over \(k\) if \(\mcl\) is very ample over \(k\): there is an immersion \(\iota:X\to\PP_k^n\) with \(\iota^*\OO(1)\cong\mcl\) [@Har10a, Chapter II, §7].
(Ample)
On a noetherian scheme \(X\), a Cartier divisor \(D\) is ample if its associated invertible sheaf \(\OO_X(D)\) is ample [@Har10a, Chapter II, §7].
If \(X\) is of finite type over \(k\), then \(D\) is ample if and only if \(nD\) is very ample over \(k\) for some \(n>0\) [@Har10a, Theorem II.7.6]. If \(X\) is a projective integral variety over an algebraically closed field, then \(D\) is very ample if and only if its complete linear system is base-point free and its associated morphism is a closed immersion. Equivalently, \(D\) is linearly equivalent to a hyperplane section under some projective embedding [@Har10a, Chapter II, §7].
(Tensor products and finite pullback)
On a noetherian scheme, the tensor product of an ample invertible sheaf with a globally generated invertible sheaf is ample, as is the tensor product of two ample invertible sheaves. If \(\mcl\) is ample and \(\mcm\) is any invertible sheaf, then \(\mcm\otimes\mcl^{\otimes n}\) is ample for all sufficiently large \(n\) [@Har10a, Exercise II.7.5]. For a finite morphism \(f:Y\to X\) of noetherian schemes and an ample invertible sheaf \(\mcl\) on \(X\), the pullback \(f^*\mcl\) is ample. Indeed, for a coherent sheaf \(\mathcal F\) on \(Y\), the sheaf \(f_*\mathcal F\) is coherent by P-AGH255PUSHCOH. For large \(n\), pull back the global generators of \(f_*\mathcal F\otimes\mcl^{\otimes n}\) and use the surjection \(f^*f_*\mathcal F\to\mathcal F\) to generate \(\mathcal F\otimes(f^*\mcl)^{\otimes n}\). On \(f^{-1}(\Spec A)=\Spec B\), with \(\mathcal F\) represented by a finite \(B\)-module \(M\), this surjection is \(B\otimes_A M\to M\), \(b\otimes m\mapsto bm\).
(An arbitrary twist need not remain ample)
On \(\PP_k^1\), the sheaf \(\OO(1)\) is ample, but \(\OO(1)\otimes\OO(-2)\cong\OO(-1)\) is not. Every positive tensor power of \(\OO(-1)\) has no global sections, so it fails the ample global-generation condition even with the coherent sheaf \(\OO_{\PP^1}\) [@Har10a, Proposition II.5.13].
(Degree on an elliptic curve)
On a smooth projective genus-one curve over an algebraically closed field, an invertible sheaf of degree one is ample and has \(h^0=1\), so its complete linear system does not give an embedding. An invertible sheaf on this curve is very ample exactly when its degree is at least three; degree three embeds it as a plane cubic [@Har10a, Chapter IV, §§1 and 3].
(Hyperplane section)
For a projective integral variety \(X\subseteq\PP_k^n\) and a hyperplane \(H\) not containing \(X\), the hyperplane section \(X\cap H\) is the effective Cartier divisor cut out by the restricted linear form. Its associated invertible sheaf is \(\OO_X(1)\). Over an algebraically closed field, the hyperplane sections form the linear system \(\PP V\subseteq\abs{\OO_X(1)}\), where \(V\) is the image of \(H^0(\PP_k^n,\OO(1))\to H^0(X,\OO_X(1))\) [@Har10a, Chapter II, §7].
Dependencies
T-DIVMAPPNThe morphism to projective space attached to a linear systemuses
Backlinks
PR-DIVAMPCURVEAmpleness of a divisor on a curveusesD-DIVEQUIVLinear, algebraic, and numerical equivalenceusesD-DIVPOLARPolarizationsusesT-DIVSERRESerre's criterion for amplenessusesFE-DIVHYPSYSThe linear system of hyperplane sectionsuses